Related papers: Subresultants and the Shape Lemma
For $a,b \ge 1$, Hilbert found in 1886 a collection of polynomial equations that cut out set-theoretically the variety X parametrizing a-th powers of binary forms of degree b. We determine the ideal of all polynomials vanishing on X,…
The projection onto the intersection of sets generally does not allow for a closed form even when the individual projection operators have explicit descriptions. In this work, we systematically analyze the projection onto the intersection…
Although intersection homology lacks a ring structure, certain expressions (called uniform) in the intersection homology of an irreducible projective variety $X$ always give the same value, when computed via the decomposition theorem on any…
We give a description of the minimal primes of the ideal generated by the 2 x 2 adjacent minors of a generic matrix. We also compute the complete prime decomposition of the ideal of adjacent m x m minors of an m x n generic matrix when the…
The concept of representing a polytope that is associated with some combinatorial optimization problem as a linear projection of a higher-dimensional polyhedron has recently received increasing attention. In this paper (written for the…
Intersection types have been originally developed as an extension of simple types, but they can also be used for refining simple types. In this survey we concentrate on the latter option; more precisely, on the use of intersection types for…
In this paper we produce the first known formula for cohomologies of the derived tensor products of structure sheaves of subschemes in the case where the intersection of the subschemes is not a local complete intersection. The case covered…
The classical Hermite-Biehler theorem describes possible zero sets of complex linear combinations of two real polynomials whose zeros strictly interlace. We provide the full characterization of zero sets for the case when this interlacing…
We introduce a new multiplication for the polytope algebra, defined via the intersection of polytopes. After establishing the foundational properties of this intersection product, we investigate finite-dimensional subalgebras that arise…
We develop a theory of sesquilinear forms over finite fields, investigating their representations via polynomials and coefficient matrices, along with classification results for these forms. Through their connection to quadratic forms, we…
We advocate the use of implicit fields for learning generative models of shapes and introduce an implicit field decoder, called IM-NET, for shape generation, aimed at improving the visual quality of the generated shapes. An implicit field…
A Poisson geometry arising from maximal commutative subalgebras is studied. A spectral sequence convergent to Hochschild homology with coefficients in a bimodule is presented. It depends on the choice of a maximal commutative subalgebra…
Complete intersections may be unexpectedly simple over fields of positive characteristic: for instance, they may be unirational despite being of general type. One explanation is given by profiles, structure that tracks the special shape of…
After introducing the definitions of positive, negative and companion rules, from a given pair of companion rules we construct a new rule with higher degree of precision The scheme is generalized giving rise to a transformation which we…
We develop a method that we call \emph{omission of intervals}, for establishing topological properties of subsets of the real line based on their combinatorial structure. Using this method, we obtain conceptual proofs of the fundamental…
Subresultant of two univariate polynomials is a fundamental object in computational algebra and geometry with many applications (for instance, parametric GCD and parametric multiplicity of roots). In this paper, we generalize the theory of…
A continuum description of unstructured meshes in two dimensions, both for planar and curved surface domains, is proposed. The meshes described are those which, in the limit of an increasingly finer mesh (smaller cells), and away from…
Differential resultant formulas are defined, for a system $\mathcal{P}$ of $n$ ordinary Laurent differential polynomials in $n-1$ differential variables. These are determinants of coefficient matrices of an extended system of polynomials…
In the present paper, we give a full description of the jet schemes of the polynomial ideal $\left( x_1\ldots x_n \right) \in k[x_1, \ldots, x_n]$ over a field of zero characteristic. We use this description to answer questions about…
In this paper, we provide constructions to enumerate large numbers of CI-liaison classes. To this end, we introduce a liaison invariant and prove several results concerning it, notably that it commutes with hypersurface sections. This…